I am searching something similar to the first order loss function ( $\mathbb{E}[max({y_{i}-y_{i},0})])$ (where $y_{i}$ is normally distributed) but for $\mathbb{E}[min(y_{i},d_{i})]$ , where $y_{i}$ is my number of products and $d_{i}$ the normally distributed random variable. I need this to maximize my expected yield of product i, which is given by $\mathbb{E}[min(y_{i},d_{i})] \cdot e_{i}$, where $e_{i}$ is the revenue of product i.
Does anyone has an idea or literature recommendations to calculate/ approximate $\mathbb{E}[min(y_{i},d_{i})]$ ?
Kind regards, Alex :)
Let $Y\sim N(0,1)$ and $d$ is a number. Then \begin{align} \mathsf{E}[Y\wedge d]&=\mathsf{E}[(Y\wedge d) 1\{Y>d\}]+\mathsf{E}[(Y\wedge d)1\{Y\le d\}] \\ &=d\,\mathsf{P}(Y>d)+\mathsf{E}[Y1\{Y\le d\}] \\ &=d(1-\Phi(d))+\int_{-\infty}^dy\phi(y)\,dy \\ &=d(1-\Phi(d))-\phi(d), \end{align} where $\Phi(\cdot)$ and $\phi(\cdot)$ are the standard normal cdf and pdf.